Day 53 – Proof of congruent triangles

Introduction We understand that a rigid motion would not change the shape or the size of the pre-image after the transformation. As a result, a rigid motion results in an image that is congruent to the pre- image. We have so far understood how we can identify corresponding angles and sides of a triangle. We now proceed further to proof discuss and the conclusion two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. We will use the concept of rigid motion.

Vocabulary Corresponding Sides Two sides are corresponding if one can be traced to back to the position of the other through one or a series of rigid motions. Corresponding Angles Two angles are corresponding if one can be traced to back to the position of the other through one or a series of rigid motions.

We would like to proof the results We would like to proof the results. Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. We will use congruence in terms of rigid motion. First, we remind ourselves that an image as a result of a rigid motion is congruent to its pre- image.

A B C 𝐶’ 𝐵’ 𝐴’ Let two triangles be congruent, and then one can be mapped onto another by one or more rigid motions such that they overlap each other. In such a case, corresponding angles and sides will be equal since rigid motion (a transformation that they went through) preserves angle and length.

Second, let two images have equal corresponding sides and angles Second, let two images have equal corresponding sides and angles. We are given that X𝑍=𝑅𝑀, 𝑍𝑌=𝑀𝑇 and 𝑌𝑋=𝑇𝑅. We also have that ∠𝑋=∠𝑅,∠𝑍=∠𝑀 and ∠𝑌=∠𝑇. T Y Z R X M

Since ∠𝑋=∠𝑅,∠𝑍=∠𝑀 and ∠𝑌=∠𝑇, we have corresponding angles equal hence this a dilation of any scale factor. T Y Z R X M

Since ∠𝑋=∠𝑅,∠𝑍=∠𝑀 and ∠𝑌=∠𝑇, we have corresponding angles equal hence this a dilation of any scale factor. We also have that X𝑍=𝑅𝑀, 𝑍𝑌=𝑀𝑇 and 𝑌𝑋=𝑇𝑅, showing that X𝑍 𝑅𝑀 =1, 𝑍𝑌 𝑀𝑇 =1 and 𝑌𝑋 𝑇𝑅 =1 Hence 𝑋𝑍 𝑅𝑀 = 𝑍𝑌 𝑀𝑇 = 𝑌𝑋 𝑇𝑅 =1 implying that a transformation of scale factor 1 exists between the two objects. Combining this result with the fact that corresponding angles are equal, we conclude that the transformation is a rigid motion. Since images and object under rigid motion are congruent, we have triangle XYZ and RMT congruent.

Example Show that the following triangles are congruent Example Show that the following triangles are congruent. Solutions Let compare the corresponding angles ∠𝑅=∠𝑃=56° ∠𝑊=∠𝑁=86° ∠𝑀=∠𝑌=38° R Y W N M P 86° 56° 38° 5 3 4.2

Thus, corresponding angles are equal Thus, corresponding angles are equal. We now compare the corresponding sides 𝑊𝑅=𝑃𝑁=3, 𝑅𝑌=𝑃𝑀=5 and 𝑌𝑊=𝑀𝑁=4.2 Thus corresponding sides are equal. By the above theorem, we have that the triangles are congruent.

homework Find out why the triangles are not congruent. Q 4.5 C N 101° B A C 101° 47° 32° 6 4 4.5

Answers to homework Corresponding angles are equal but sides are not

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